What I am trying to do is find a way to email a shared mailbox and have it create an entry for the calendar on the Web application Office 365. The whole outcome is to share a calendar with a group to see who it's out sick. I am just trying to figure out an easy way to keep tabs of this and have a way to push out a visual report to the group and see who is out. Any help would be great!

If this is not about sending out meeting requests, you could look into MS Flow. Think of it as MS macro like programming / automation where you set up conditions and actions. It can be used for most o365 services - Exchange, SharePoint, Planner etc. I have created a flow for my own mailbox that creates Outlook task depending on sender and mail subject.

I was thinking the same thing Jones! But should I start from scratch or find a template and build from there? If I build from a template, which would be a good one to start with? Thank you so much for your help!

It should not be too complicated, think you can just start from scratch. There might be ready templates, but I can't check that now. All you do is log on to https://flow.microsoft.com with shared mailbox account (you probably will have to enable it first if you disable your shared mbx accounts). What I am not sure is whether you need to keep account enabled for created flow to trigger

All that said Adam-H's suggestion is better/easier if you can just set all this up so that meeting invites / appointments are sent. Then shared mbx will work as if it's a room mailbox

Another alternative might be to set up o365 group (they also have mailbox/calendar). Group owners could easily add members themselves without bothering admins

There is also MS Staffhub which might be even the best solution. It's a workshift scheduler. Instead of noting when which person has a shift, you just mark when someone is absent. Not sure if it integrates well in Outlook (there might be addons), but there is website and also mobile app.